Staring was an expression.
History
When Matthew Corbett took Sooty and Sweep of The Sooty Show to the seaside with Iris Wildthyme sometime in the late 20th century, after it began to rain, Matthew said to the audience that if Sooty wasn't orange, he would've thought that Sooty had been replaced by Paddington to give him such a stare. (PROSE: It's Raining Gin [+]Jon Arnold, The Curse of Fanfic! (The Curse of Fanfic!, Obverse Books, 2020). Page 136.)
When Edward J Green wished the Ninth Doctor a "good morning" in a newsagents while buying a copy of Heat in early March 2005, the Doctor just stared back at Edward. (PROSE: Have You Seen This Man? [+]various authors, Who is Doctor Who? (BBC, 2005).)
When Giles crossed a road, talking on his smart phone, he stopped on the other side next to a white van to stare at Lucifer. (WC: Mission: Find Lilith [+]Trevor Spencer, Hellscape (BBV Productions, TikTok and YouTube, 2023).)
Behind the scenes
Information from invalid sources
Whilst in a bar, the Cyber Man once aggressively asked the Second Doctor "Who you starin' at?". (NOTVALID: Doctor Who? 204)